Matrix Norm
Given a square complex or real matrix , a matrix norm
is a nonnegative number associated with
having the properties
1. when
and
iff
,
2. for any scalar
,
3. ,
4. .
Let , ...,
be the eigenvalues of
, then
![]() | (1) |
The matrix -norm is defined for a real number
and a matrix
by
![]() | (2) |
where is a vector norm. The task of computing a matrix
-norm is difficult for
since it is a nonlinear optimization problem with constraints.
Matrix norms are implemented as Norm[m, p], where may be 1, 2, Infinity, or "Frobenius".
The maximum absolute column sum norm is defined as
![]() | (3) |
The spectral norm , which is the square root of the maximum eigenvalue of
(where
is the conjugate transpose),
![]() | (4) |
is often referred to as "the" matrix norm.
The maximum absolute row sum norm is defined by
![]() | (5) |
,
, and
satisfy the inequality
![]() |